Question Bank on Determinants & Matrix

ffiIIEWI(c,S TAR.GET IIT JEE 2OI3

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& MATRICES

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a lsl

lu' \,/

The value of the deterrninant

lcos{nx1

lsin(nx)

(c)

(B) a

ttR= (A)

1

[r

rl liLr il^l anddet.

(D)

x

a,

n and x

1-In,neNhenthevalueof l,is (c) 3

Ais an iirvotutary mabix given ry

o=

Io

r'l

I

(c)tA

A-r

(B)

A

-3 4 | tnen the inverse of ; Ll -3 4)

z

lf a, 0 & y are real numbers, then

I t O

=

(D)A2

cos(p-cr)

lcos(cr-P)

I

cos(y cos(T

lcos(o-1) cos(P-y) (A) -1 (C) cosry + cosP + cosl

(B)

matric*

[;

lj[l?][;

then the value of n is (A) 26

I

16.

Ji

(c)-qr 3l U

Il

F

i Lx y zj

iAldet{8)

(c) 377

(D) 378

=

-

and supposethatdet.(A)=2thenthedet.(B)equals, where

Zar

2a -pl 2b -ql 2c -rl

2

(D) det(B) = $

\/l

z-^dl-'l

2ap I Zaq I (A) A., = A,

g= [1,. | 4Y 14,

(B)det1a1=-g

2

detiB) = - 16

fr 37sl [o tJ

l-l rl is "qr"tto the matrix

io qh .;l r

letn=l

=

cosc cosp

lt|.lj$ ^

4r

lf the product of n

-a) -P)

zeto

,,o=[-3;$ ;l'Jr-],A-1 sgvenby (A)-A

will be

P

KJ Lr

1

(B) A, =

2At

\=2d--.

lf an idempotent matrix is also skew symmetric then it must be (B) an identi$ matrix (A) an involutiary matrix a nullmatrix. (C) an orthogonal matrix

dr

(D)4 +24=0

flo PflI

'o

Vibri

Let the matri

(A)2

Let f(x) =

Ir |

I

()2

+

wino st gtenlents is not conect

-g

+Ar

ftfo ,.6

(D) (kA)r = kAr (k is a scalar)

,'oll yqf

of a 3 x 3 matrixA= (a,,) is

defin follo

"9 fil

)l-1

?r+

dzz+ aee (i.e. T,(A) is sum of the main

ways hold?

1n; t, (kA)= kT.(A)(k is a scalar) (c) T,(t.)= 3

,

is not coi'rect

(D)

nalelements). Which of the

rr(A+ts)=T..(A)+Tn(B) T,(A2) = (T,(A) )2

Put )t=o (A)

16.

'r]

(B)(Ar)r=A

lf A and B are invertible matrices, which one of the

lr ftt tu_

|

(A)Adj.4=lAlA-1 .r$XA + B)-1 = B-1 + A-1

=lJ -t -_ -\ n g-{

' fl

33

(B) 0

lf Aand B are difiere'rt matrices satisfying As = (A) det (A2 + 82) must be zero. (B) det (A - B) must be zerc. (C) det (A2 + 82) as well as det (A - B) must be (D) At least one of det (A2 + 82) or det (A- B) must be zero.

The system of equations x + v + z = 5, x + 2y + 32 = 9, x + 3y + ).2 =

(A)1,=5,p=13

p

has no solution if

(B)1,=5 (D)p=13

tfll.=S,p+i3

I

a.t 0 clfl'4' r.- r -l p=

(B) a2 + 621 sz (D) ab+bc+ca ect of two square matrices A and B conformable for sum and product.

(B)tr(cA)=ot(A),aeR

(A) 0 (C) independentofQ Vibrant Academy (I) Pvt Ltd.'A-14(A)"

i

ofO

of

0&Q

both

Road tlo.1.Indraprastha IndustrialArea, Kota (Raj.) Ph.O744-242340G

t3l

A' ( | A Which of the following statements is incorrect for a square matrix (A) lf A is a diagonal matrix, A-1 will also be.a diagonal T_?J'I_^-,..

| + 0)

fr

(/

dr2 ert

I

lutt Let Ao = art azz dzt | lut. atz att

of Ao and and let A" denote the determinant formed by the cofactors of elements

4

A^ denote the determinant formed by the cofactor at at An-., then the determinant value of An is

and so on An denotes the determinant formed by the

c6factors

tn

(A)

(D)

(c)

A2o'

^i;

^1

^1

o sin a cos cr I t l- cos2 p sin B cos p I lf A= ,ino.oro sin2cr | I ' = lsinp cosp sin2p i [

.or'

integral ,r"ln that, AB is a null matrix, then which of the following should necessarily be an odd

"r"

(D)c+P

(B) p

cos

The deterrninant I cos I

sin

gq a"n cL

(x-y) cos (Y-z) (x+y)

cos (Y + z)

(x+y)

sin (Y+z)

(A) 2 sin (x - Y) sin (Y - z) sin (z - x) (C) 2 cos (x - Y) cos (Y - z) cos (z- x)

')''

2 sin (x - y) sin (y - z) sin (z-

x)

*-'->

:

2 cos (x - Y) cos (Y- z) cos (z- x)

I coso -ti1e ragivenmatrixA=l sine cosu I I

wnicn of the following statement holds good?

(B)A is symmetric, for 0 = (2n + 1)

(A)A=A-1 V0eR

1l

, ' nel

(D)Ais a skew symmetric, for0 = nr ; n e

B,(A) (1 +a+b+c) (1 +x+V+z)

R,rRr+

,{

I

/\

- 3(ax+by+cz)

+yz+ zx\ tx+ y) + b (y + z) + c(z+ x) - (xy (ab+ bc+ca) iCi x(a * 6) * y(b+ c) + z(c+ a) -

tei i

(D) none oftheEe lf A is matrix such

th

+

A+ 2l = O) tnen which of the following is INCORRECT ?

(A)Ais non-singular (B)A

1

ere

t

IO

is unit matrix of order

Vibrant Academy (I) hrt. Ltd.

'A

2

1 \J%n-'= ; {n+ l)

and O is null matrix of order 2

(D)A-1

)

A+

o.'1, Indraprastha IndustrialArea, Kota (Raj.) Ph'

It

I'

(A+ l)

&ft-l

0744-242uo6

=Q t4l

Mbra

Let A+2'.= then Tr (A) (A) 0

-

(B) a

*

[B

lt t) (D) none

2

1

+bl

.*91 a-ol

ad

,o-'=

sl

Tr(B) has the value equalto

rts of Ao and

rmed bythe

+ f] .,,0

[_L

lz -r

and D, =

bc, is

aca+clD, b d bld I tn"n the value of ID2 a c a+b+cl

wnere b

*

0 and

(c)-2b

(B) 0

The number of solutions of the matrix equation X2 =l other than I, is (c)2 (B) 1 (A) (where I is the 2 x 2 unit matrix )

0

(l+b2)x (l+c2)x l+b2x (l + c2)x (l + b2)x l+c2x

dd integral

(A)

0

(B)

o,u"no=

(A)i'e0

[l t] ;l = [l

1

?]

'to-r.risasinsurarmatrixthen

(C)1,2+32,,+4=0 (D)).,2-3i,-6=0

,%"-3X-4=0 Thevaluesof0,l.forwhichthefollowingequations n A sinOx-cos0y+(I+1)z=0; cos0x+sin0Y- :=0; l.x+(l+1)y+@s0z=0 U -" L' have non trivial solution, is

R-t0) (G)e=(2n+1)ll,1. eR*,nel (A)0= Dn,l. e

I t

el Let A=

=l

G)g=2nn,L isanyrationalnumber n

=(2n+1);,1eR,nel

sino 1l

I sinol,whereoso